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Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectralcharacterization of an isomorphism from a $C^\star$-algebra onto a Banachalgebra. We then use this result to show that a $C^\star$-algebra $A$ isisomorphic to a Banach algebra $B$ if and only if there exists a surjectivefunction $\phi:A\rightarrow B$ satisfying (i)$\sigma\left(\phi(x)\phi(y)\phi(z)\right)=\sigma\left(xyz\right)$ for all$x,y,z\in A$ (where $\sigma$ denotes the spectrum), and (ii) $\phi$ iscontinuous at $\mathbf 1$. A simple example shows that (i) cannot be relaxed toproducts of two elements, as is the case with commutative Banach algebras. Ourresults also elaborate on a paper ([3]) of Bre\v{s}ar and \v{S}penko.

A Spectral Characterization of Isomorphisms on $C^\star$-Algebras